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Locally Hausdorff tight groupoids generalised

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Abstract

We extend Exel’s ample tight groupoid construction to non-ample groupoids, even in the general locally Hausdorff case.

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References

  1. Bice, T.: Distance domains: Completion. Topology Proc. 54, 7–57 (2019). http://www.topology.auburn.edu/tp/reprints/v54/tp54002p1.pdf

  2. Bice, T., Starling, C.: General non-commutative locally compact locally Hausdorff Stone duality. Adv. Math. 314, 40–91 (2019). https://doi.org/10.1016/j.aim.2018.10.031

    Article  MathSciNet  MATH  Google Scholar 

  3. Bice, T., Starling, C.: Hausdorff tight groupoids generalised. Semigroup Forum 100(2), 399–438 (2020). https://doi.org/10.1007/s00233-019-10027-y

    Article  MathSciNet  MATH  Google Scholar 

  4. Exel, R.: Inverse semigroups and combinatorial \(C^\ast \)-algebras. Bull. Braz. Math. Soc. (N.S.) 39(2), 191–313 (2008). https://doi.org/10.1007/s00574-008-0080-7

    Article  MathSciNet  MATH  Google Scholar 

  5. Exel, R.: Reconstructing a totally disconnected groupoid from its ample semigroup. Proc. Amer. Math. Soc. 138(8), 2991–3001 (2010). https://doi.org/10.1090/S0002-9939-10-10346-3

    Article  MathSciNet  MATH  Google Scholar 

  6. Exel, R., Pardo, E.: The tight groupoid of an inverse semigroup. Semigroup Forum 92(1), 274–303 (2016). https://doi.org/10.1007/s00233-015-9758-5

    Article  MathSciNet  MATH  Google Scholar 

  7. Kunen, K.: Set Theory, Studies in Logic (London), vol. 34. College Publications, London (2011)

    Google Scholar 

  8. Lawson, M.V.: Inverse Semigroups. The Theory of Partial Symmetries. World Scientific Publishing Co. Inc., River Edge, NJ (1998). https://doi.org/10.1142/9789812816689

    Book  MATH  Google Scholar 

  9. Lawson, M.V.: Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras. Int. J. Algebra Comput. 22(06), 1250058 (2012). https://doi.org/10.1142/S0218196712500580

    Article  MathSciNet  MATH  Google Scholar 

  10. Lawson, M.V., Lenz, D.H.: Pseudogroups and their étale groupoids. Adv. Math. 244(Supplement C), 117–170 (2013). https://doi.org/10.1016/j.aim.2013.04.022

    Article  MathSciNet  MATH  Google Scholar 

  11. Lawson, M.V., Margolis, S.W., Steinberg, B.: The étale groupoid of an inverse semigroup as a groupoid of filters. J. Aust. Math. Soc. 94(2), 234–256 (2013). https://doi.org/10.1017/S144678871200050X

    Article  MathSciNet  MATH  Google Scholar 

  12. Lenz, D.H.: On an order-based construction of a topological groupoid from an inverse semigroup. Proc. Edinb. Math. Soc. (2) 51(2), 387–406 (2008). https://doi.org/10.1017/S0013091506000083

    Article  MathSciNet  MATH  Google Scholar 

  13. Resende, P.: Étale groupoids and their quantales. Adv. Math. 208(1), 147–209 (2007). https://doi.org/10.1016/j.aim.2006.02.004

    Article  MathSciNet  MATH  Google Scholar 

  14. Shirota, T.: A generalization of a theorem of I. Kaplansky. Osaka Math. J. 4, 121–132 (1952). http://projecteuclid.org/euclid.ojm/1200687806

  15. Steinberg, B.: A groupoid approach to discrete inverse semigroup algebras. Adv. Math. 223(2), 689–727 (2010). https://doi.org/10.1016/j.aim.2009.09.001

    Article  MathSciNet  MATH  Google Scholar 

  16. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40(1), 37–111 (1936). https://doi.org/10.2307/1989664

    Article  MathSciNet  MATH  Google Scholar 

  17. Vries, H.D.: Compact spaces and compactifications: An algebraic approach. Thesis Amsterdam (1962)

  18. Wallman, H.: Lattices and topological spaces. Ann. Math. (2) 39(1), 112–126 (1938). https://doi.org/10.2307/1968717

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, Prague, Czech Republic

    Tristan Bice

  2. Carleton University, Ottawa, Canada

    Charles Starling

Authors
  1. Tristan Bice
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  2. Charles Starling
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Correspondence to Tristan Bice.

Additional information

Communicated by Mark V. Lawson.

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Tristan Bice is supported by the GAČR project EXPRO 20-31529X and RVO: 67985840. Charles Starling is supported by a Carleton University internal research grant.

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Bice, T., Starling, C. Locally Hausdorff tight groupoids generalised. Semigroup Forum 102, 583–611 (2021). https://doi.org/10.1007/s00233-020-10149-8

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  • DOI: https://doi.org/10.1007/s00233-020-10149-8

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